Binary Decision Diagram with Minimum Expected Path Length

Transcription

1 Binary Decision Diagram with Minimum Expected Path Length Yi-Yu Liu Kuo-Hua Wang TingTing Hwang C. L. Liu Department of Computer Science, National Tsing Hua University, Hsinchu 300, Taiwan Dept. of Computer Science and Information Engineering, Fu Jen Catholic University, Taipei, Taiwan Abstract We present methods to generate a Binary Decision Diagram (BDD) with minimum expected A BDD is a generic data structure which is widely used in several fields. One important application is the representation of Boolean functions. A BDD representation enables us to evaluate a Boolean function: Simply traverse the BDD from the root to the terminal and retrieve the value in the terminal. For a BDD with minimum expected path length will be also minimized the evaluation time for the corresponding Boolean function. Three efficient algorithms for constructing BDDs with minimum expected path length are proposed. 1. Introduction Binary Decision Diagram (BDD) was introduced in 1978 [1] as an effective way to represent Boolean functions that leads to efficient manipulation [2] and implementation [3]. Given a random input pattern, we can traverse the BDD to determine the outputing. There are previous works on minimizing the number of s in a BDD [4] [5] [6]. However, the number of s is not directly related to the evaluation time for a Boolean function. Rather, the evaluation time is directly related to the total expected path length of a BDD. In a logic circuit, the probabilities of occurance of the input variables are different. It follows that the evaluation time for different input combinations are also different. Consequently, a BDD with minimum expected path length will also have minimum evaluation time. To obtain a BDD with minimum expected path length, we need to develop (1) methods for computing the expected path length of a BDD (2) heuristics for ordering the variables in a BDD to minimize the expected This paper is organized as follows. Besides Section 1 being an introduction, we will discuss basic concept of BDDs in Section 2. In Section 3, we will propose three methods to compute the expected path length of an ROBDD. Section 4 deals with heuristics to determine variable ordering so that the resultant ROBDD will have minimum expected Finally, experiment results will be presented. 2. Preliminary 2.1. ROBDD A BDD is a directed, acyclic graph which consists of s and edges. There are two types of s, terminal and non-terminal. A terminal is a Boolean value or, and a non-terminal is an input Boolean variable. Each non-terminal contains two outgoing edges pointing to other s, which is referred to as the parent of these s. For convenience, we distinguish the two edges as then-edge and else-edge. A which is connected from a then-edge is the then-child of its parent- and from an else-edge the else-child. A Reduced Ordered BDD (ROBDD) [2] is a BDD with a chosen variable ordering in all paths from the root to the terminal s and possesses the following properties: (1) if both the then-child and else-child of a non-terminal are identical, the non-terminal will be removed from the path, and (2) if there are two or more identical subtrees, only one will be retained and all parent-s which have the same subtree will share one subtree as their children. The formula representation of an ROBDD is if parent then then-child else else-child, which is called an ITE operator (IF-THEN-ELSE). By using the ITE operator, Boolean operations can be easily performed on ROBDDs. Figure 1 gives an ROBDD for the function with the input variable ordering. Note that for the same function, different ordering of input variables will result in different ROBDDs Expected Path Length of ROBDD Since the evaluation time of a ROBDD is directly related to the path length of ROBDD, we want to shorten the path lengths with high probability, i.e., to minimize the expected path length of the ROBDD. It is necessary that we be able to compute a expected path length of a given ROBDD. To begin with, we define the expected length of the th path in ROBDD as, where is the probability of the th path and is the path length of the th path. The total Expected Path Length (EPL) of a ROBDD rooted at is defined as (1) /2001/$ IEEE

2 F A B BEPL (a) a C 1- P a P a 0 1 BEPL (a ) 0 a 0 a 1 BEPL (a 1) Figure 1. An ROBDD example. where is the number of paths of the ROBDD. Following this formula, to calculate the expected path length will need directly enumerate all paths. This approach is impractical because the number of paths may grows exponentially. In the following section, we will propose three efficient methods to calculate expected length: bottom-up method, topdown method, and middle-way method. 3. Three Methods for Computing EPL Since it is impractical to enumerate all paths in a ROBDD. We present three efficient methods in this section for computing the EPL of a ROBDD. They are bottom-up method, top-down method, and middle-way method. The first and second methods need to be executed only once over all s. After initializing, the third method will be activated to improve the performace of our algorithm Bottom-up Method Bottom-up method is formulated by which the expected path length can be calculated from terminal s up to root. Since at each non-terminal, the expected path lengths of its two child-s are already computed, we can compute the expected path length of the non-terminal based on the expected path lengths for its child-s. Before we proceed to develop the formula, we define the following notations as: [ ] the expected path length of all paths from to terminal s [ ] the probability of the th path from to a terminal [ ] the path length of the th path from to a terminal Now, let be a non-terminal with its then-child and else-child as shown in Figure 2. The expected path lengths of all paths from terminal s up to and can supposedly be computed as (2) (3) Figure 2. Bottom-up method for calculating the expected where and are the total number of paths from terminal s up to and, respectively. Let the probability from to be. Then, the probability from to will be. The expected path length from terminal s up to s is computed as Note that the total path probability from a non-terminal to terminal s is. That is, From the identity of equations (2) (3) (5), we can reduce equation (4) to From equation (6), we follow that the expected path length of a non-terminal at can be computed by following steps : step1 multiply the probability of the then-child by the expected path length of the then-child which is computed in the previous stage and is readily used. step2 multiply the probability of the else-child by the expected path length of the else-child which is computed in the previous stage and is readily used. step3 sum results of step1, step2, and constant 1 (5) (6) (4) 2 709

3 With boundary condition that both the path length of terminal s and are 0, we can recursively calculate the expected length of the root of an ROBDD by bottom-up method. root 3.2. Top-down Method Reversely, top-down method is used to calculate the expected path length from root down to terminal s. Since at each non-terminal, the expected path lengths of its parent s are already computed, we can compute the expected path length of an non-terminal based on the expected path lengths for its parent s. Before we proceed to develop the formula, we define the following notations as: [ ] the expected path length of all paths from root to non-terminal [ ] the probability of the th path from root to [ ] the sum of probabilities of all paths from root to [ ] the path length of the th path from root to Obviously, from the above definitions, equations (7) (8) hold. That is, the summation of probabilities of all paths from root to a is (7) where is the total number of paths from root to. Moreover, the expected path length from root to a is (8) Now, let be a non-terminal with its parent-s labeled to and the probabilities from parent s to to be to, respectively, as shown in Figure 3. We are to compute the expected path length of all paths from root to. First, the summation of probabilities of all paths from root to is (9) where is the number of parent-s of. Now, the expected path length from root to is (10) By equations (7) (8) and (9), equation (10) can be reduced to (11) From equation (11), we follow that the expected path length of a non-terminal at can be computed by following steps: step1 for all parent s, multiply the probability of parent- to the by the expected path length from root to the parent which is computed in the previous stage and is readily used. b 1 b j TEPL p b j b b j TEPL b Figure 3. Top-down method for calculating the expected step2 sum probabilities of all paths from root to by equation (9), where the sum of probabilities of all path from root to a parent- is already computed and readily used. step3 sum results of step1 and step2 Based on the assumptions that the path length of root is and the probabilities from all paths to root is 1, we can calculate the expected length of the terminal s of an ROBDD by top down method. Finally, the expected path length of the whole ROBDD is the summation of the expected path lengths of the two terminal s Middle-way Method The third method is used to compute the expected path length of an ROBDD assuming that the expected path lengths of all non-terminal s at level have been computed by top-down method and the expected path lengths of all non-terminal s at level have been computed by bottom-up method. Let be the cut between levels and and there are edges crossing that connect the s at levels and as shown in Figure 4. To compute the expected path length of the whole ROBDD, we have to consider all paths connected by edges. First, we consider how to compute the expected path length connected by a single crossing edge. In this case, edge connects the top end- and the bottom end- as shown in Figure 4. Suppose that the probability of edge be and there are paths from root to, and paths from terminals s to as shown in Figure 4. Then, the expected path length of all paths,, which pass through edge is b n 3 710

4 TEPL(b ) b k k p k edge k a k BEPL (a k) C Algorithm find-least-cost( ) Input: Boolean function; Output: return and ; Begin (1) find an initial variable ordering for ROBDD of ; (2) build an initial ROBDD; (3) compute the cost of the ROBDD; (4) while ( not stop ) do (5) find new variable ordering; (6) restructure the ROBDD according to the new variable ordering; (7) compute the cost of the new ROBDD; (8) update the cost function; (9) check if the stopping criterion is set; endwhile (10) return and ; End Figure 4. Middle-way method for calculating the expected By equations (2)(5)(8)(9), equation (12) can be reduced to (12) (13) From equation (13), we follow that the expected path length of all paths passing through edge connecting and can be computed by following steps: step1 sum the expected path length from root of all paths to, the expected path length of all paths from to terminals and twice of the total path probability to. All of them are already computed and are readily used. step2 multiply the result of step1 and the probability of edge Finally, the total expected length of the whole ROBDD is 4. Algorithms for Finding Low Cost ROBDD Our algorithm begins with an initial ordering. Then, an iterative loop is entered to modify the initial ordering. In each iteration, a new ordering is determined and the old ROBDD is restructured based on the new variable ordering. Next, the cost for the new ROBDD is computed. The loop continues till the stopping criterion is met. The procedure is shown in Figure 5. The detailed description is explained as follows. In step 1, an initial ordering needs to be determined to construct an initial ROBDD. It can be determined by constructing a minimum size ROBDD. In this case, algorithms Figure 5. The find-least-cost algorithm proposed in [5] [6] can be used. Step 3 is to compute the cost of the initial ROBDD. For the computation of the expected path length of the ROBDD, bottom-up and top-down methods proposed in Section 3 are used and all computed data for each is stored at the. Step 5 finds a new input ordering. Window permutation algorithm [5] [6] which finds a local minimum in a window of size and sifting algorithm [4] which looks for a suitable position for each variable in each iteration can be used. With the new input variable ordering, step 6 restructures the old ROBDD to a new ROBDD. Transposition operator [7] is used which restructures an ROBDD by several simple ROBDD operations. For example, if the order of and are swapped in a new ordering, the following formula can be used. Note that swapping is not limited to adjacent variables. In step 7, the cost for the new ROBDD is computed. For the expected length of the new ROBDD, similar to step 6, it needs not be computed from scratch. Instead, we will use the information for the old ROBDD to calculate the new expected length of ROBDD. Suppose that we have a new ordering that variables at levels and are swapped as shown in Figure 6. Since the part of ROBDD above level and the part of ROBDD below level will not be changed as seen in Figure 6, we do not have to recompute them. We only need to compute the new expected path length for the s in the middle. On the one side, the bottom-up method calculates the expected path length from s at level up. On the other side, the top-down method calculates the expected path length from s at level down. When variables on the two sides meet in the middle, the middleway method can be used. By using middle-way method, it prevents much more redundant calculations than top-down and bottom-up methods. 5. Experimental Results Our experiment is performed on a SUN-Ultra Enterprise 150. Software platform is based on the ROBDD package in 4 711

5 STP TEPL BEPL invalid level i level j Figure 6. Computation for parts lying between the two swapped variables. SIS [8]. LGSynth93 benchmark suite is used in our experiment. Experiment is conducted to find a minimum expected path length ROBDD. First, the sifting algorithm [4] is used to find a minimum-sized ROBDD which can be used as our initial ROBDD. The reason for this initial ordering heuristic is that a small size ROBDD may lead to a small expected length ROBDD. In subsequent steps, window permutation is used to generate a new input ordering and the expected path length for the new ROBDD is computed. Finally, the probabilities of input variables are set to be equal. Table 1 shows the results. Column gives the results of ROBDD with initial ordering whose objective is to find an ROBDD with minimum size. Column gives the results of ROBDD with expected path length as objective. Columns and give the expected path length and size of ROBDDs, respectively. In column ratio, we compute the ratio of the results of minimum s to the results of minimum expected It is clear from Table 1 that in most cases, the expected path length can be reduced at the cost of small increase of the number of s except cordic and f51m. In some cases (con1, vg2, cm150a, etc.), we found that both the size and the expected path length of ROBDD are reduced. On the average, the expected path length is reduced by 25% while the the number of s is increased by 10%. 6. Conclusions In this work, we present methods to generate a ROBDD with minimized expected With smaller EPL of a ROBDD, the evaluation time of a Boolean function will be reduced. Three methods are proposed to calculate the expected path length of a ROBDD efficiently. The experimental results indicate that the expected path length of an ROBDD can be reduced at the cost of a small increase of OBDD size. Table 1. Results of expected path length with equal input probability circuit min min exp path ratio (%) path size path size path size 5xp alu b con cordic ex inc sao vg misex cm150a cm151a cm162a cm163a cm85a mux z4ml f51m pcle traffic Average References [1] S. B. Akers, Binary decision diagrams, IEEE Trans. Comput., Vol. C-27, pp , June [2] R. E. Bryant, Graph-based algorithms for boolean function manipulation, IEEE Trans. on Comput., Vol. C-35, No. 8, August [3] K. S. Brace, R. L. Rudell, and R. E. Bryant, Efficient Implementation of a BDD Package, DAC, [4] R. Rudell, Dynamic variables ordering for ordered binary decision diagrams, ICCAD, [5] M. Fujita, Y. Matsunaga, and T. Kakuda, On Variable Ordering of Binary Decision Diagrams for the Application of Multi-level Logic Synthesis, EDAC, pp 50-54, Mar [6] N. Ishiura, H. Sawada, and S. Yajima, Minimization of Binary Decision Diagrams Based on Exchanges of Variables ICCAD, pp , Nov [7] K. H. Wang, T. T. Hwang, and C. Chen, Restructuring Binary Decision Diagrams Based on Functional Equivalence, EDAC, pp , Feb [8] E. M. Sentovich, K. J. Singh, L. Lavagno, C. Moon, R. Murgai, A. Saldanha, H. Savoj, P. R. Stephan, R. K. Brayton, A. Sangiovanni-Vincentelli, SIS: A System for Sequential Circuit Synthesis, Electronics Research Laboratory Memorandum No. UCB/ERL M92/41, 4 May